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Question

Harold plays a game in which he starts with $2. Each game has 2 rounds; in each round, the amount of money he starts the round with is randomly either added to or multiplied by a number, which is randomly either 1 or 0. The choice of arithmetic operation and of number are independent of each other and from round to round. If Harold plays the two-round game repeatedly, the long-run average amount of money he is left with at the end of the game, per game, is between

Answer

First, decode the game. In each round, you either add 1, multiply by 1, add 0, or multiply by 0. Each of those possibilities is equally likely. Now, notice that multiplying by 1 and adding 0 do the same thing: they leave the number unchanged. So, in effect, each round has 3 possible outcomes:

Add 1 (25% or 1/4 chance)

Leave the number unchanged (50% or 1/2 chance)

Multiply by 0, turning the number to 0 (25% or 1/4 chance)

Let’s now trace the game through. Harold starts with $2. After the first round, Harold either has $3 (1/4 chance), $2 (1/2 chance), or $0 (1/4 chance). Now figure out the second round from each starting point:

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